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STATISTICS &
 PROBABILITY

Hypothesis Testing
What is a Hypothesis?
I assume the mean    • an assumption about
 GPA of this class     the population
      is 3.5!
                       parameter
             • an educated guess
               about the population
               parameter
Hypotheses Testing: This is the process of
  making an inference or generalization on
population parameters based on the results of
            the study on samples.
                  Reject?
                  Accept?




    Statistical Hypotheses: It is a guess or
        prediction made by the researcher
   regarding the possible outcome of the study.
Hypotheses Testing
is deciding between what is
          REALITY
          and what
     is COINCIDENCE!
Types of Statistical Hypotheses
       Null Hypothesis (Ho): is
       always hoped to be rejected
       Always contains “=“ sign
Alternative Hypothesis (Ha):
•Challenges Ho
•Never contains “=“ sign
•Uses “< or > or ≠ “
•It generally represents the idea
 which the researcher wants to prove.
The Null Hypothesis: Ho
Ex. Ho: The average GPA of this class is 3.5
             µ = 3.5
             H0:
     The Alternative Hypothesis: Ha

     Ha: The average GPA of this class is
             a) higher than 3.5 (Ha: µ > 3.5)
             b) lower than 3.5 (Ha: µ < 3.5)
             c) not equal to 3.5 (Ha:
µ ≠ 3.5)
Types of Hypotheses Tests
1. One-tailed left directional test
   – this is used if Ha uses < symbol


                              Critical value is
α = 0.05                          obtained
                Acceptance     from the table
                  region
           Area = 0.05
   Rejection region
Types of Hypotheses Tests
 2. One-tailed right directional test
    – this is used if Ha uses > symbol


                              Critical value is
α = 0.05     Acceptance
                                  obtained
                               from the table
               region
                   Area= 0.05
                      Rejection region
Types of Hypotheses Tests
3. Two-tailed test: Non-directional
   – this is used if Ha uses ≠ symbol


                                Critical value is
 α = 0.05/2                         obtained
               Acceptance
                                 from the table
                 region

           Area=.025   Area=.025
   Rejection region     Rejection region
Level of Significance, α and the Rejection Region

α 0.05,
 =        means the probability of being right is 95% , and
 the probability of being wrong is 5%. So what is α 0.01?
                                                    =




                                          Rejection region
                  Acceptance              Area is 0.05
                    Region

                                  α = 0.05
                  .
Level of Significance, α and the Rejection Region

α=   0.01, means the researcher is taking a 1% risk of being
 wrong and a 99% risk of being right. So, what is α = 0.05?




 Rejection region
 Area is 0.01     Acceptance
                   Region


         α = 0.01
Level of Significance, α and the Rejection Region

α = 0.05 means the probability of committing Type I error is 5%.

     α = 0.05, since it is 2-T, then α = 0.05/2= 0.025


                       Acceptance
                         region


                 Area=.025     Area=.025
         Rejection region       Rejection region
Level of Significance, α and the Rejection
                         Region
To summarize:
α
= 0.05,  means the probability of being right is 95% and the
 probability of being wrong is 5%. So what is α 0.01?
                                                 =


α=    0.01, means the researcher is taking a 1% risk of being
 wrong and a 99% risk of being right. So, what is α = 0.05?

α = 0.05 means the probability of committing Type I error is
5%.
                   α
      So what is       = 0.01?
Errors in Hypothesis Testing


                                           Ho: ERAP is not guilty
Type I (α error )         Errors
Rejecting a true                             If the court convicts
      Ho!                                    ERAP, when in fact
                                              he is not guilty, the
                    Errors in Decisions
                                                 court commits
                                                  Type I error!

                Errors in Conclusions

   Type I is the same as the       α or the level of significance.
                                                  © 1984-1994 T/Maker Co.
Errors in Hypothesis Testing


Type II (β error )     Errors
Accepting a false                      Ho: ERAP is not guilty
      Ho!
                                           If the court acquits
                 Errors in Decisions       ERAP, when in fact
                                             he is guilty, the
                                              court commits
                                              Type II error!
                 Errors in Conclusions
Decisions made regarding Ho
 (Reject Ho/Do not reject Ho)

If we reject Ho, it means it is wrong!


       If we do not reject Ho,
    it doesn’t mean it is correct,
          we just don’t have
           enough evidence
              to reject it!


                                 © 1984-1994 T/Maker Co.
Testing the Significance of Difference Between Means


      Z-test
                         σ   is known
      n ≥ 30

     t-test
                        σ is unknown
     n < 30

     F-test
                         3 or more µ s
   (ANOVA)
Testing the Significance of Difference Between Means
       “n is large or when n ≥ 30 & σ is known.”

              Z-test               σ   is known
              n ≥ 30
      • Hypothesized/population mean VS Sample mean
       and population standard deviation is known.


       Z=
          (x − µ )    n     x - is the sample mean
                            µ - is the population mean
                 σ          n - is the sample size
                           σ - is the population std. dev.
              Using PHStat: Go to…
“One-Sample Tests; Z-Test for the Mean:Sigma Known”
Testing the Significance of Difference Between Means
      “n is large or when n ≥ 30 & σ is known.”
           Z-test
                                σ   is known
           n ≥ 30
         • Sample mean 1 VS Sample mean 2
      and population standard deviation is known.


        x1 − x 2          x1 - is the mean of sample 1
   Z=                     x 2 - is the mean of sample 2
          1 1
      σ     +        n1 & n2 - are the sample
          n1 n2           σ - is the population std. dev.
                             sizes
Testing the Significance of Difference Between Means
      “n is large or when n ≥ 30 & σ is known.”
              Z-test
                                  σ is unknown
              n ≥ 30
           • Sample mean 1 VS Sample mean 2
       and 2 sample standard deviations are known.

                            x1 - is the mean of sample 1
       x1 − x 2
 Z=                         x 2 - is the mean of sample 2
       s12 s2 2 n1 & n2 - are the sample
          +
       n1 n2 s1 & s2 - are the sample std. devs.
                        sizes
Using Microsoft Excel: Go to…“Z-Test: Two-Sample For Means”
The Critical Value Approach in
        Testing the Significance of Difference
                    Between Means
The 5-step solution
Step 1. Formulate Ho and Ha

Step 2. Set the level of significance α , usually
        it is given in the problem.
Step 3. Formulate the decision rule (when to reject Ho);
        Find the critical value/P-value.
Step 4. Make your decision.

Step 5. Formulate your conclusion.
Approaches in
                      Hypothesis Testing

                  Critical value    p- value
                    approach        approach

      Computed vs. Critical                      p-valueα vs.α
                                                    α ≤
5-step solution                    5-step solution
1.Ho: ___________                  1.Ho : ___________
   Ha: ___________                   Ha : ___________
2.α = ___; Cri-value= ______       2. α = ___; p- value=________
                                       α
3. Decision rule: Reject Ho        3. Decision rule: Reject Ho if
 if Comp − value ≥ Cri − value       p- value ≤ α
                                               ≤α
4. Decision:     α                 4. Decision:
5. Conclusion:                     5. Conclusion:
Finding Critical Values: One-Tailed
    What Is Z Given α = 0.05?


       .45
             .05
  Α= α = .05
                      Z         4     5
       Z=1.65
     Critical value   1.5   .4382   .4394
                      1.6   .4495   .4505
Critical Values: Z - Table

                  α         .01            0.05
       Type
       One-T            ±   2.33        ± 1.65
       Two-T            ±   2.58       ±   1.96

You will refer to this table to get the critical value of Z
or the Z tabular .
CRITERION:
1. One-tailed test (right directional)
   “Reject H0 if Zc ≥ Zt “
2. One-tailed test (left directional)
   “Reject H0 if Zc ≤ Zt
3. Two-tailed test (Zc = +)
    “Reject H0 if Zc ≥ Zt “
4. Two-tailed test (Zc = -)
    ‘Reject H0 if Zc ≤ Zt “
EXERCISES:
1. Past records showed that the average
 final examination grade of students in
 Statistics was 70 with standard deviation
 of 8.0. A random sample of 100 students
 was taken and found to have a mean final
 examination grade of 71.8. Is this an
 indication that the sample grade is better
 than the rest of the students? Test at 0.05
 level of significance.
2. A certain type of battery is known to have
 a mean life of 60 hours. In random sample
 of 40 batteries, the mean life was found to
 be 58 hours with a standard deviation of
 4.5 hours. Does it indicate that the mean
 lifetime of such battery has been reduced?
 Test at 0.01 level of significance.
3. The manager of a rent-a-car business
  wants to know whether the true average
  numbers of cars rented a day is 25 with a
  standard deviation of 6.9 rentals. A
  random sample of 30 days was taken and
  found to have an average of 22.8 rentals.
  Is there a significance between the mean
  and the sample mean? Test at 0.05 level
  of significance.
4. Advertisements claim that the average
 nicotine content of a certain kind of
 cigarette is 0.30 milligram. Suspecting that
 this figure is too low, a consumer
 protection service takes a random sample
 of 50 of these cigarette from different
 production slots and find that their nicotine
 content has a mean of 0.33 milligram with
 a standard deviation of 0.18 milligram.
 Use the 0.05 level of significance to test
 the null hypothesis µ = 0.30 against the
 alternative hypothesis µ < 0.30.
5. An experiment was planned to compare the mean time
   (in days) required to recover from common cold for
   person given a daily doze of 4 mgs. of vitamin C versus
   those who were not given a vitamin supplement.
   Suppose that 35 adults were randomly selected for each
   treatment category and that the mean recovery times
   and standard deviations for the 2 groups were as
   follows:
                     n         X         δ
   W/ vit. C        35        5.8       1.2
   W/o vit. C       35        6.9       5.8
  Suppose your research objective is to show that the use
   of vit. C increases the mean time required to recover
   from common cold. Test using α = 0.05.

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Statistics Presentation week 7

  • 2. What is a Hypothesis? I assume the mean • an assumption about GPA of this class the population is 3.5! parameter • an educated guess about the population parameter
  • 3. Hypotheses Testing: This is the process of making an inference or generalization on population parameters based on the results of the study on samples. Reject? Accept? Statistical Hypotheses: It is a guess or prediction made by the researcher regarding the possible outcome of the study.
  • 4. Hypotheses Testing is deciding between what is REALITY and what is COINCIDENCE!
  • 5. Types of Statistical Hypotheses Null Hypothesis (Ho): is always hoped to be rejected Always contains “=“ sign Alternative Hypothesis (Ha): •Challenges Ho •Never contains “=“ sign •Uses “< or > or ≠ “ •It generally represents the idea which the researcher wants to prove.
  • 6. The Null Hypothesis: Ho Ex. Ho: The average GPA of this class is 3.5 µ = 3.5 H0: The Alternative Hypothesis: Ha Ha: The average GPA of this class is a) higher than 3.5 (Ha: µ > 3.5) b) lower than 3.5 (Ha: µ < 3.5) c) not equal to 3.5 (Ha: µ ≠ 3.5)
  • 7. Types of Hypotheses Tests 1. One-tailed left directional test – this is used if Ha uses < symbol Critical value is α = 0.05 obtained Acceptance from the table region Area = 0.05 Rejection region
  • 8. Types of Hypotheses Tests 2. One-tailed right directional test – this is used if Ha uses > symbol Critical value is α = 0.05 Acceptance obtained from the table region Area= 0.05 Rejection region
  • 9. Types of Hypotheses Tests 3. Two-tailed test: Non-directional – this is used if Ha uses ≠ symbol Critical value is α = 0.05/2 obtained Acceptance from the table region Area=.025 Area=.025 Rejection region Rejection region
  • 10. Level of Significance, α and the Rejection Region α 0.05, = means the probability of being right is 95% , and the probability of being wrong is 5%. So what is α 0.01? = Rejection region Acceptance Area is 0.05 Region α = 0.05 .
  • 11. Level of Significance, α and the Rejection Region α= 0.01, means the researcher is taking a 1% risk of being wrong and a 99% risk of being right. So, what is α = 0.05? Rejection region Area is 0.01 Acceptance Region α = 0.01
  • 12. Level of Significance, α and the Rejection Region α = 0.05 means the probability of committing Type I error is 5%. α = 0.05, since it is 2-T, then α = 0.05/2= 0.025 Acceptance region Area=.025 Area=.025 Rejection region Rejection region
  • 13. Level of Significance, α and the Rejection Region To summarize: α = 0.05, means the probability of being right is 95% and the probability of being wrong is 5%. So what is α 0.01? = α= 0.01, means the researcher is taking a 1% risk of being wrong and a 99% risk of being right. So, what is α = 0.05? α = 0.05 means the probability of committing Type I error is 5%. α So what is = 0.01?
  • 14. Errors in Hypothesis Testing Ho: ERAP is not guilty Type I (α error ) Errors Rejecting a true If the court convicts Ho! ERAP, when in fact he is not guilty, the Errors in Decisions court commits Type I error! Errors in Conclusions Type I is the same as the α or the level of significance. © 1984-1994 T/Maker Co.
  • 15. Errors in Hypothesis Testing Type II (β error ) Errors Accepting a false Ho: ERAP is not guilty Ho! If the court acquits Errors in Decisions ERAP, when in fact he is guilty, the court commits Type II error! Errors in Conclusions
  • 16. Decisions made regarding Ho (Reject Ho/Do not reject Ho) If we reject Ho, it means it is wrong! If we do not reject Ho, it doesn’t mean it is correct, we just don’t have enough evidence to reject it! © 1984-1994 T/Maker Co.
  • 17. Testing the Significance of Difference Between Means Z-test σ is known n ≥ 30 t-test σ is unknown n < 30 F-test 3 or more µ s (ANOVA)
  • 18. Testing the Significance of Difference Between Means “n is large or when n ≥ 30 & σ is known.” Z-test σ is known n ≥ 30 • Hypothesized/population mean VS Sample mean and population standard deviation is known. Z= (x − µ ) n x - is the sample mean µ - is the population mean σ n - is the sample size σ - is the population std. dev. Using PHStat: Go to… “One-Sample Tests; Z-Test for the Mean:Sigma Known”
  • 19. Testing the Significance of Difference Between Means “n is large or when n ≥ 30 & σ is known.” Z-test σ is known n ≥ 30 • Sample mean 1 VS Sample mean 2 and population standard deviation is known. x1 − x 2 x1 - is the mean of sample 1 Z= x 2 - is the mean of sample 2 1 1 σ + n1 & n2 - are the sample n1 n2 σ - is the population std. dev. sizes
  • 20. Testing the Significance of Difference Between Means “n is large or when n ≥ 30 & σ is known.” Z-test σ is unknown n ≥ 30 • Sample mean 1 VS Sample mean 2 and 2 sample standard deviations are known. x1 - is the mean of sample 1 x1 − x 2 Z= x 2 - is the mean of sample 2 s12 s2 2 n1 & n2 - are the sample + n1 n2 s1 & s2 - are the sample std. devs. sizes Using Microsoft Excel: Go to…“Z-Test: Two-Sample For Means”
  • 21. The Critical Value Approach in Testing the Significance of Difference Between Means The 5-step solution Step 1. Formulate Ho and Ha Step 2. Set the level of significance α , usually it is given in the problem. Step 3. Formulate the decision rule (when to reject Ho); Find the critical value/P-value. Step 4. Make your decision. Step 5. Formulate your conclusion.
  • 22. Approaches in Hypothesis Testing Critical value p- value approach approach Computed vs. Critical p-valueα vs.α α ≤ 5-step solution 5-step solution 1.Ho: ___________ 1.Ho : ___________ Ha: ___________ Ha : ___________ 2.α = ___; Cri-value= ______ 2. α = ___; p- value=________ α 3. Decision rule: Reject Ho 3. Decision rule: Reject Ho if if Comp − value ≥ Cri − value p- value ≤ α ≤α 4. Decision: α 4. Decision: 5. Conclusion: 5. Conclusion:
  • 23. Finding Critical Values: One-Tailed What Is Z Given α = 0.05? .45 .05 Α= α = .05 Z 4 5 Z=1.65 Critical value 1.5 .4382 .4394 1.6 .4495 .4505
  • 24. Critical Values: Z - Table α .01 0.05 Type One-T ± 2.33 ± 1.65 Two-T ± 2.58 ± 1.96 You will refer to this table to get the critical value of Z or the Z tabular .
  • 25. CRITERION: 1. One-tailed test (right directional) “Reject H0 if Zc ≥ Zt “ 2. One-tailed test (left directional) “Reject H0 if Zc ≤ Zt 3. Two-tailed test (Zc = +) “Reject H0 if Zc ≥ Zt “ 4. Two-tailed test (Zc = -) ‘Reject H0 if Zc ≤ Zt “
  • 26. EXERCISES: 1. Past records showed that the average final examination grade of students in Statistics was 70 with standard deviation of 8.0. A random sample of 100 students was taken and found to have a mean final examination grade of 71.8. Is this an indication that the sample grade is better than the rest of the students? Test at 0.05 level of significance.
  • 27. 2. A certain type of battery is known to have a mean life of 60 hours. In random sample of 40 batteries, the mean life was found to be 58 hours with a standard deviation of 4.5 hours. Does it indicate that the mean lifetime of such battery has been reduced? Test at 0.01 level of significance.
  • 28. 3. The manager of a rent-a-car business wants to know whether the true average numbers of cars rented a day is 25 with a standard deviation of 6.9 rentals. A random sample of 30 days was taken and found to have an average of 22.8 rentals. Is there a significance between the mean and the sample mean? Test at 0.05 level of significance.
  • 29. 4. Advertisements claim that the average nicotine content of a certain kind of cigarette is 0.30 milligram. Suspecting that this figure is too low, a consumer protection service takes a random sample of 50 of these cigarette from different production slots and find that their nicotine content has a mean of 0.33 milligram with a standard deviation of 0.18 milligram. Use the 0.05 level of significance to test the null hypothesis µ = 0.30 against the alternative hypothesis µ < 0.30.
  • 30. 5. An experiment was planned to compare the mean time (in days) required to recover from common cold for person given a daily doze of 4 mgs. of vitamin C versus those who were not given a vitamin supplement. Suppose that 35 adults were randomly selected for each treatment category and that the mean recovery times and standard deviations for the 2 groups were as follows: n X δ W/ vit. C 35 5.8 1.2 W/o vit. C 35 6.9 5.8 Suppose your research objective is to show that the use of vit. C increases the mean time required to recover from common cold. Test using α = 0.05.